thermodynamic properties of the shastry-sutherland model
TRANSCRIPT
PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase
Alexander Wietek ,1,2,* Philippe Corboz,3 Stefan Wessel,4 B. Normand,5 Frédéric Mila,6 and Andreas Honecker7
1Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA2Institut für Theoretische Physik, Universität Innsbruck, 6020 Innsbruck, Austria
3Institute for Theoretical Physics and Delta Institute for Theoretical Physics, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, Netherlands
4Institut für Theoretische Festkörperphysik, JARA-FIT, and JARA-HPC, RWTH Aachen University, 52056 Aachen, Germany5Neutrons and Muons Research Division, Paul Scherrer Institute, CH-5232 Villigen-PSI, Switzerland
6Institute of Physics, Ecole Polytechnique Fédérale Lausanne (EPFL), CH-1015 Lausanne, Switzerland7Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, Université de Cergy-Pontoise,
95302 Cergy-Pontoise Cedex, France
(Received 4 July 2019; revised manuscript received 11 September 2019; published 21 October 2019)
The thermodynamic properties of the Shastry-Sutherland model have posed one of the longest-lastingconundrums in frustrated quantum magnetism. Over a wide range on both sides of the quantum phase transition(QPT) from the dimer-product state to the plaquette-based ground state, neither analytical nor any availablenumerical methods have come close to reproducing the physics of the excited states and thermal response. Wesolve this problem in the dimer-product phase by introducing two qualitative advances in computational physics.One is the use of thermal pure quantum (TPQ) states to augment dramatically the size of clusters amenableto exact diagonalization. The second is the use of tensor-network methods, in the form of infinite projectedentangled-pair states (iPEPS), for the calculation of finite-temperature quantities. We demonstrate convergenceas a function of system size in TPQ calculations and of bond dimension in our iPEPS results, with completemutual agreement even extremely close to the QPT. Our methods reveal a remarkably sharp and low-lyingfeature in the magnetic specific heat, whose origin appears to lie in a proliferation of excitations composed oftwo-triplon bound states. The surprisingly low energy scale and apparently extended spatial nature of these statesexplain the failure of less refined numerical approaches to capture their physics. Both of our methods will havebroad and immediate application in addressing the thermodynamic response of a wide range of highly frustratedmagnetic models and materials.
DOI: 10.1103/PhysRevResearch.1.033038
I. INTRODUCTION
Frustrated quantum spin systems, particularly those for-bidding magnetically ordered ground states, provide one ofthe most important avenues in condensed matter physicsfor the realization and investigation of phenomena rangingfrom fractionalization to many-particle bound states, frommassive degeneracy to topological order and from quantumentanglement to quantum criticality [1]. Significant progresshas been made in describing the ground states of many suchquantum magnets in two and three dimensions, includingdifferent types of valence-bond crystal and both gapped andgapless quantum spin liquids [2]. However, a full understand-ing of the excitation spectrum and thermodynamic responseof frustrated quantum spin models, which is the key to ex-perimental interpretation, has remained elusive in all but a
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small number of special cases. This is because no unbiasedanalytical or numerical methods exist by which to study theseproperties for general systems at low but finite temperaturesand on lattices whose sizes approximate the thermodynamiclimit.
A case in point is the Shastry-Sutherland model [3], whichwas constructed explicitly to have as its ground state a productstate of dimer singlets. Also referred to as the orthogonaldimer model [4], it is defined by the Hamiltonian
H = JD
∑〈i, j〉
�Si · �S j + J∑〈〈i, j〉〉
�Si · �S j, (1)
which is represented schematically in Fig. 1. Here JD isan intradimer coupling (denoted by 〈i j〉) and J a mutuallyfrustrating interdimer coupling (denoted by 〈〈i j〉〉). The groundstate is clearly a dimer-product phase at small J/JD and asquare-lattice antiferromagnet at large coupling ratios. As in-timated above, the ground-state phase diagram is in fact quitewell known, with the most quantitatively accurate results pro-vided by the ansatz of infinite projected entangled-pair states(iPEPS), which we will extend here to finite temperatures [5].As shown in Fig. 1, the singlet dimer-product state is foundup to J/JD = 0.675(2) and the antiferromagnetic state beyond
2643-1564/2019/1(3)/033038(20) 033038-1 Published by the American Physical Society
ALEXANDER WIETEK et al. PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
JD
J/JD
J
0.675(2) 0.765(15)
dimer plaquette Néel
FIG. 1. Geometry and coupling constants, JD and J (centerpanel), of the Shastry-Sutherland model [Eq. (1)]. The three panelsshow schematic representations of the singlet dimer-product state,which is exact, the plaquette state, and the Néel phase. Shaded el-lipses and squares denote, respectively, dimer- and plaquette-singletstates. Phase boundaries are taken from Ref. [5].
J/JD = 0.765(15), with a gapped plaquette-based state occur-ring in the intermediate regime. The lower quantum phasetransition (QPT) is of first order; the upper one has also beenfound numerically to be first order, but only weakly so [5],and this has also been questioned [6]. Any finite temperaturewill destroy the magnetic order of the antiferromagnetic phasein this two-dimensional (2D) system. By contrast, in theplaquette phase one expects an Ising transition to occur atfinite temperatures as the plaquette order, which breaks thelattice symmetry, is lost. In the dimer-product state, no suchphysics is expected.
Somewhat remarkably, the Shastry-Sutherland model hasa faithful realization in the material SrCu2(BO3)2, which hasexactly the right geometry and only weak non-Heisenberg(primarily Dzyaloshinskii-Moriya) interactions. Discoveredin 1991 [7] and first studied for its magnetic properties20 years ago this year [8], SrCu2(BO3)2 lies in the dimer-product phase [9] and shows a very flat band of elementarytriplet (“triplon”) excitations [10,11]. Excellent magnetic sus-ceptibility [8,12] and specific-heat [13] measurements madeat this time were used to estimate the coupling ratio asJ/JD � 0.635, thereby placing SrCu2(BO3)2 rather close tothe dimer-plaquette QPT. Detailed experiments performedin the intervening two decades have revealed a spectacularseries of magnetization plateaus [8,14–21] as well as a curiousredistribution of spectral weight at temperatures very low onthe scale of the triplon gap [11,22]. Of most interest to ourcurrent study is the result [23] that an applied pressure makesit possible to increase the ratio J/JD to the extent that, atapproximately 1.9 GPa, the material is pushed through theQPT into a plaquette phase. Data for the specific heat underpressure have appeared concurrently with the present study[24] and indicate that the low-temperature peak moves to alower temperature at 1.1 GPa before evidence of an orderedphase appears at 1.8 GPa.
In contrast to the ground state, the excited states andthermodynamics of the Shastry-Sutherland model are not atall well known around the QPT, despite the attention focusedon this regime due to SrCu2(BO3)2. Full exact diagonalization(ED) has been limited to systems of up to 20 sites and, as wewill demonstrate in detail here, suffer from significant finite-size limitations at J/JD > 0.6. Quantum Monte Carlo (QMC)
methods suffer from extreme sign problems, and despite arecent study by some of us that expands the boundaries ofwhat is possible by QMC [25], the low-temperature behaviorin the regime J/JD > 0.6 remains entirely inaccessible. iPEPSmethods work in the thermodynamic limit and are immune tofrustration problems, but to date have been limited to ground-state properties. As a consequence, two decades after theirmeasurement, the best interpretation of the thermodynamicproperties of SrCu2(BO3)2 [12,13], and the origin of theestimate J/JD = 0.635, remain based on ED of 16- and 20-siteclusters [4,26].
Here we introduce two qualitative technical advances innumerical methods for computing thermodynamic proper-ties and benchmark them by application to the Shastry-Sutherland model in the dimer-product phase up to the QPT.For a quantum many-body system, a thermal equilibriumstate can be represented by a typical pure state, which isknown as a thermal pure quantum (TPQ) state [27,28].This result can be understood as a consequence of a moregeneral phenomenon, known as quantum typicality [29,30].All statistical-mechanical quantities can be obtained fromaveraging over a few TPQ states. For practical purposes,obtaining TPQ states by ED allows access to thermody-namic quantities without having to perform a full diagonal-ization, and hence admits a qualitative increase in acces-sible system sizes [31,32]. We will exploit TPQ methodsto reach system sizes of N = 40 in the Shastry-Sutherlandproblem.
iPEPS are a tensor-network ansatz to represent a quantumstate on an infinite lattice [33–35], here in two dimensions,whose accuracy is controlled by the bond dimension D of thetensor. In fact, this ansatz provides a compact representationnot only for the ground states of gapped local Hamiltonians,but also for representing a purification of the thermal densityoperator, and hence the thermal states of local Hamiltonians.Here we exploit recently developed algorithms [36] allow-ing the efficient calculation of imaginary-time evolution pro-cesses, which in general require additional truncation to retainthe value of D at each time step, to generate thermal states ofthe Shastry-Sutherland Hamiltonian and hence to compute itsthermodynamic properties.
Our TPQ and finite-T iPEPS methods enable us to cap-ture the physics of the Shastry-Sutherland model, and thusof SrCu2(BO3)2, in a way not hitherto possible near theQPT. To illustrate this clearly, in Fig. 2 we show sampleresults from both methods for the magnetic specific heats andsusceptibilities of Shastry-Sutherland models with couplingratios J/JD = 0.60, 0.63, 0.65, and 0.66. Over this smallrange of couplings near the QPT, the specific heat developsa pronounced two-peak form, with a broad higher peak lyingjust beyond the range of Fig. 2(a). The narrow lower peak,presumably also characteristic of finite-energy excitations inthe magnetically disordered dimer-product phase, becomessurprisingly sharp and moves to a remarkably low energy asthe QPT is approached. The corresponding susceptibilitiesrise increasingly abruptly before an increasingly “square”turnover at their maximum. We draw attention to the factthat the results from both our methods are in excellent quan-titative agreement, apart from the specific-heat peak heightsextremely close to the QPT. These characteristic shapes, and
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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35T/JD
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
C
(a)
TPQ N = 36
J/JD = 0.60, R = 20
J/JD = 0.63, R = 40
J/JD = 0.65, R = 20
J/JD = 0.66, R = 20
iPEPS D = 18
J/JD = 0.60
J/JD = 0.63
J/JD = 0.65
J/JD = 0.66
D = 18
J/JDJ = 0.60
J/JDJ = 0.63
J/JDJ = 0.65
J/JDJ = 0.66
iPEPS D = 18
J/JD = 0.60
J/JD = 0.63
J/JD = 0.65
J/JD = 0.66
0.00 0.05 0.10 0.15 0.20 0.25 0.30T/JD
0.00
0.05
0.10
0.15
0.20
0.25
0.30
χJ
D
(b)
TPQ N = 36
J/JD = 0.60, R = 20
J/JD = 0.63, R = 40
J/JD = 0.65, R = 20
J/JD = 0.66, R = 20
D = 18
J/JD = 0.60
J/JD = 0.63
J/JD = 0.65
J/JD = 0.66
D = 18
J/JDJ = 0.60
J/JDJ = 0.63
J/JDJ = 0.65
J/JDJ = 0.66
D = 18
J/JD = 0.60
J/JD = 0.63
J/JD = 0.65
J/JD = 0.66
iPEPS
FIG. 2. (a) Magnetic specific heat C(T ) and (b) magnetic sus-ceptibility χ (T ) of the Shastry-Sutherland model in the regime ofcoupling ratios 0.60 � J/JD � 0.66. The number of sites in the TPQcalculations is N = 36 and the shaded regions show the standarderror of the TPQ method for the respective coupling ratios (Sec. IV).The iPEPS bond dimension is D = 18.
their systematic evolution, have not been obtained before andshed light both on the thermodynamic quantities themselvesand on the underlying excitation spectrum responsible fortheir form.
The paper is organized as follows. In Sec. II we introduceTPQ states and their application to the calculation of thermo-dynamic quantities. In Sec. III we provide a brief introductionto iPEPS and discuss the generation of thermal states by theirimaginary-time evolution. Section IV presents our completeresults from both methods for the magnetic specific heatand susceptibility of the Shastry-Sutherland model over thecoupling range 0.60 � J/JD � 0.66, which are summarizedin Fig. 2. In Sec. V we discuss the physics of the sharpfeatures we observe in the specific heat near the QPT, thelimits of our numerical methods in the context of the Shastry-Sutherland model, and the application of our results to the ma-terial SrCu2(BO3)2. Section VI contains a brief summary andperspective concerning the impact of our results in frustratedquantum magnetism.
II. THERMODYNAMICS FROM THERMAL PUREQUANTUM STATES
To evaluate the thermal average of a quantum mechanicalobservable A in the canonical ensemble, one computes
〈A〉 = Tr(e−βH A)/Z, (2)
where H denotes the Hamiltonian, β = 1/kBT the inversetemperature, kB the Boltzmann constant, and Z = Tr(e−βH )the canonical partition function. TPQ states provide an alter-native approach to the evaluation of thermodynamic proper-ties [27,28]. The trace of any operator A can be evaluatedby taking the average of its expectation values over a set ofrandom vectors
Tr(A) = d 〈r|A|r〉, (3)
where |r〉 denotes a normalized vector with random normal-distributed components and d denotes the dimension of theHilbert space. Hence, any thermal average can be estimatedby evaluating
〈A〉 = 〈β|A|β〉〈β|β〉 , (4)
where
|β〉 ≡ e− β
2 H |r〉 (5)
denotes the “canonical TPQ” state [28] and the overbar de-notes the average over the random vectors |r〉. Importantly, theTPQ state |β〉 is still random. The variance of the estimate inEq. (4) decreases as 1/
√d , where d = 2N for an S = 1/2 lat-
tice model on N sites, and hence becomes exponentially smallas a function of the system size under only mild assumptionsabout the form of the operator H ; a precise statement maybe found in Ref. [28]. For larger system sizes, it is thereforenecessary to generate only relatively few random TPQ states|β〉 in order to achieve small statistical errors. Because thesamples in the denominator and the numerator of Eq. (4)are correlated when evaluated from the same set of randomvectors |r〉, we perform a jackknife analysis [37] to estimatethe error bars.
The key advantage of using TPQ states, when comparedwith evaluating statistical averages using the standard for-malism of Eq. (2), is that the state |β〉 can be approximatednumerically to arbitrary precision without resort to a fulldiagonalization of the Hamiltonian H . Variants of the TPQmethod have been employed in several numerical studiesof static and dynamical observables [31,32,38–42]. WhereasRef. [28] proposed a Taylor-expansion technique, here weimprove the speed of convergence by using the Lanczos basis[43] of the Krylov space of H to evaluate the states |β〉.
The nth orthonormal basis vector in the Lanczos basis |vn〉is given recursively [44] by
|vn〉 = |vn〉 /bn, (6)with
|vn〉 = H |vn−1〉 − an−1 |vn−1〉 − bn−1 |vn−2〉 , (7)
where |v1〉 = |r〉, b1 = 0, and
an = 〈vn|H |vn〉 , bn = ‖vn‖. (8)
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ALEXANDER WIETEK et al. PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
By defining the matrix of Lanczos vectors
Vn = (v1| · · · |vn), (9)
the nth Lanczos approximation of H can be written as
H ≈ VnTnV†
n , (10)
where Tn is a tridiagonal matrix given by
Tn =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
a1 b2 0 · · · 0
b2 a2 b3 0...
0 b3. . .
. . . 0...
. . . an−1 bn
0 · · · 0 bn an
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
. (11)
We use this approximation to evaluate
〈β|A|β〉 = 〈r|e− β
2 H Ae− β
2 H |r〉≈ 〈r|Vne− β
2 TnV †n AVne− β
2 TnV †n |r〉
= e1e− β
2 TnV †n AVne− β
2 Tn e1, (12)
where e1 = (1, 0, . . . , 0)T. In the case that A is a power of theHamiltonian A = Hk , we have
V †n HkVn = T k
n . (13)
We stress that the only quantities to be computed are thesequences an and bn. Further, results at all temperatures canbe obtained from a single evaluation of the Lanczos basis. Theexponentiated tridiagonal matrices e− β
2 Tn are computed by theeigendecomposition of the matrices Tn and exponentiation ofthe diagonal matrix of eigenvalues. The eigendecompositionof Tn is computed with the LAPACK routine dsteqr [45,46],which uses an implicit QR method [47–49]. The dimension nof the Krylov space is increased until the desired precision isachieved. The computations in this paper require a dimensionn < 150 to achieve a precision superior to the statistical errors.
In our computations we use a block-diagonal form of theHamiltonian
H =⊕
ρ
Hρ, (14)
where the blocks Hρ correspond to the Hamiltonian in differ-ent lattice-symmetry and total-Sz sectors. Statistical averagesare evaluated in each individual block and then summed,
Tr(e−βH A) =∑
ρ
Tr(e−βHρ Aρ ). (15)
We employ the techniques proposed in Ref. [50] to performthe computations in the individual symmetry sectors.
It has recently become possible to compute Lanczos ap-proximations in particular symmetry sectors for systems ofup to N = 50 S = 1/2 sites for the Heisenberg model on thesquare lattice and N = 48 S = 1/2 on the kagome lattice [50].It is worth noting here that the latter geometry and model areparticularly favorable for the TPQ method [28] because thedensity of states is high at very low energies [51]. However,deriving thermodynamic quantities from the TPQ method
requires that these Lanczos approximations be computed mul-tiple times in each symmetry sector for a number R of random“seeds,” i.e., independent random realizations of the vector|r〉 in Eq. (5), and it is R that determines the statistical error.Thus restrictions on CPU time currently limit the practicalapplication of the TPQ approach to an upper limit of 40–42S = 1/2 spins. More specifically, the calculations in this paperwere performed with R = 80 seeds for all coupling ratiosJ/JD on clusters of N = 32 spins and R = 20 for the N =36 cluster, except at J/JD = 0.63, where R = 40 seeds wereevaluated. For the N = 40 cluster we used R = 5. In total, theTPQ calculations in this paper required approximately 5.5 MCPU hours.
We comment that our implementation of the TPQ methodis reminiscent of the finite-temperature Lanczos method [52],which has also recently been pushed to a cluster of N =42 S = 1/2 spins for the kagome lattice [53]. Although theprecise relation between these two different methods remainsto be understood, currently we believe that the TPQ ap-proach has a more systematic theoretical foundation and asa consequence that the errors within this approach are bettercontrolled.
For system sizes N � 20, full diagonalization of theHamiltonian yields numerically exact results with no statis-tical errors. An intermediate method is to compute a largenumber of low-lying eigenstates, for example by using thevariant of the Lanczos method outlined in Ref. [54], andto approximate the low-temperature thermodynamics usingthese. Despite the significant advantage of also providingresults that are entirely free of statistical errors, this methodremains restricted by the fact that the required numbers ofeigenvalues are large even at the surprisingly low temperatureswhere the dominant physical processes occur in the presentcase (Sec. IV). Thus we present benchmarking Lanczos EDresults for cluster sizes up to N = 28, although we commentthat N = 32 would be accessible were we to invest CPU timescomparable to those used in our TPQ calculations.
As an illustration of the manner in which TPQ system sizesallow access to the thermodynamic properties of the Shastry-Sutherland model, in Fig. 3(a) we show the specific heat perdimer
C(T ) ≡ 2
N
∂E
∂T= 2
Nβ2[〈H2〉 − 〈H〉2], (16)
where E ≡ 〈H〉 is the energy, obtained from our TPQ calcula-tions for coupling ratio J/JD = 0.63 with N = 32, 36, and 40.Figure 3(b) shows the corresponding magnetic susceptibilityper dimer
χ (T ) ≡ 2
N
∂m
∂h
∣∣∣∣h=0
= 2
Nβ[〈M2〉 − 〈M〉2], (17)
where h is a magnetic field applied along the z axis and m ≡〈M〉, with
M =N∑
i=1
Szi , (18)
denotes the total magnetization. In both Figs. 3(a) and 3(b), wecompare these TPQ results with ED calculations for systemsof sizes N = 16 and 20 and at low temperatures N = 24 and
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0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.0
0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
χJ
D
(b) J/JD = 0.63
ED N = 16
ED N = 20
ED N = 24
ED N = 28
TPQ N = 32, R = 80
TPQ N = 36, R = 40
TPQ N = 40, R = 5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C
(a) J/JD = 0.63
ED N = 16
ED N = 20
ED N = 24
ED N = 28
TPQ N = 32, R = 80
TPQ N = 36, R = 40
TPQ N = 40, R = 5
FIG. 3. (a) Magnetic specific heat C(T ) and (b) susceptibilityχ (T ) per dimer at coupling ratio J/JD = 0.63, comparing resultsobtained from ED and TPQ calculations for system sizes fromN = 16 to 40 sites.
28. The details and context of these results will be discussed infull in Secs. IV and V; for the purposes of this introduction, westress that both C(T ) and χ (T ) have sharp physical features,namely the low-temperature peak in the former and shoulderin the latter, whose shape is manifestly not well reproducedat small N but becomes very much clearer as N is increased.From Fig. 2, this type of access to larger N values becomesincreasingly important as the QPT is approached.
III. THERMODYNAMICS FROM AN iPEPS
An infinite projected entangled-pair state, as a tensor-network ansatz representing a 2D quantum state in the thermo-dynamic limit [33–35], can be seen as a natural generalizationof infinite matrix-product states (MPS) to two, or indeedhigher, dimensions. While the rank of the tensors used inthe representation is determined by the physical problem, theamount of information they contain is determined by theirbond dimension D, which is used to control the accuracy ofthe ansatz. Although iPEPS were introduced originally for
b p
rl
p
rl
a
B
BA
A
B
BA
A
(a) (b)
(c) (d)
(e)
t
b
t
FIG. 4. (a) Graphical representation of an iPEPS tensor Artlbp in
which each leg corresponds to an index of the tensor. The physicalindex p (green leg) represents the local Hilbert space of a latticesite (here a dimer) and the other four, r, t , l , and b, are auxiliaryindices, each with bond dimension D, which connect neighboringtensors. (b) Standard iPEPS representing a pure state, shown here fora unit cell containing two different tensors, A and B, arranged in acheckerboard pattern on the infinite lattice. A connection betweentwo tensors implies taking the sum over the corresponding index.(c) iPEPS tensor with an extra index a (blue leg) representing thelocal Hilbert space of an ancilla site. (d) iPEPS ansatz for thepurification |�(β )〉 of the thermal density operator ρ(β ). (e) Repre-sentation of ρ(β ) = Tra |�(β )〉 〈�(β )|, where the trace is taken overall ancilla degrees of freedom (shown by the connected blue lines).
representing the ground states of local Hamiltonians, morerecently several iPEPS methods have been developed for therepresentation of thermal states [36,55–65]. Here we focuson the approaches discussed in Ref. [36], where an iPEPSis used to represent a purification of the 2D thermal densityoperator. However, we note that an alternative route to thecalculation of finite-temperature properties would be by adirect contraction of the associated (2+1)D tensor networkusing (higher-order) tensor renormalization-group methods[66]. We comment that MPS-based methods have also beenapplied recently [67–70] to compute the thermal response offinite 2D systems in cylindrical geometries; these approachesprovide accurate results for cylinders up to a circumferenceW , which is limited by an exponential increase of the requiredbond dimension with W .
An iPEPS consists of a unit cell of tensors which isrepeated periodically on an infinite lattice. For a ground state(pure state), each tensor has one physical index p describingthe local Hilbert space of the dimer and four auxiliary indicesr, t , l , and b [Fig. 4(a)], each with bond dimension D, whichconnect the tensor to its four nearest neighbors on a squarelattice. For the Shastry-Sutherland model (Fig. 1) we requireone tensor per dimer, arranged in a unit cell with two tensors,A and B, in a checkerboard pattern, as shown in Fig. 4(b).
We represent a thermal density operator, ρ(β ) = e−βH , atinverse temperature β by its “purification,” |�(β )〉, whichis a pure state in an enlarged Hilbert space where eachphysical site j (here a dimer with local dimension d = 4) isaccompanied by an ancilla site with the same local dimension,described by a local basis |p, a〉 j , where p and a denote,respectively, the local physical and ancilla basis states. Therepresentation of the thermal state is through its thermaldensity operator ρ(β ) = Tra |�(β )〉〈�(β )|, where the trace is
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ALEXANDER WIETEK et al. PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
taken over all ancilla degrees of freedom a. Physically, theancilla states act as a perfect heat bath, and taking the tracegives exact thermodynamic averages. In tensor-network nota-tion, each ancilla site is incorporated through the additionalindex a [blue lines in Fig. 4(c)], leading to the diagrammaticrepresentation of |�(β )〉 shown in Fig. 4(d) and ρ(β ) inFig. 4(e).1
At infinite temperature (β = 0), |�(0)〉 can be representedby the product state |�(0)〉 = ∏
j
∑dk=1 |k, k〉 j (with bond
dimension D = 1), which for ρ(0) yields the identity operator.The purified state at any given β can be obtained as |�(β )〉 =U (β/2) |�(0)〉, where the imaginary-time evolution operatorU (β ) = e−βH acts on the physical degrees of freedom of|�(0)〉. In practice, we use a Trotter-Suzuki decompositionof U (β ), meaning it is split into M imaginary-time steps τ =β/M, and approximate the operator for a single time step by aproduct of nearest-neighbor time-evolution operators U (τ ) =∏
〈i, j〉 Ui j (τ ), with Ui j (τ ) = e−τ Hi j , where 〈i, j〉 denotes allnearest-neighbor site pairs in the unit cell.2
Each application of Ui j increases the bond dimensionbetween the tensors on sites i and j to a larger value, D′ > D,which for reasons of efficiency must be truncated back to Dat every step. As in the imaginary-time evolution of groundstates, there exist several approaches for the truncation of abond index, the two most common being the simple- [71] andfull-update methods [34,72]. In the simple-update approach,the truncation is performed using a local approximationof the state, which is computationally cheap but does notyield the optimal truncation. In the full-update approach, abond is truncated in a manner optimal under the method bywhich the entire state is taken into account; however, this isconsiderably more computationally expensive and in generalit is not possible to reach bond dimensions as large as by thesimple-update technique. Below we compare results obtainedby these two approaches. Further details concerning theiPEPS ansatz and algorithms for thermal states may be foundin Ref. [36].
For contraction of the thermal-state tensor network, whichis required to compute all physical expectation values, weuse a variant [73] of the corner-transfer-matrix (CTM)renormalization-group method [74,75] in which the accuracyis controlled by the boundary bond dimension, χ . To improvethe efficiency of the calculations, we exploit the global U(1)symmetry [76,77] of the model of Eq. (1). Further detailsof the iPEPS methods we employ here may be found inRefs. [5,72,78].
Turning to the evaluation of the thermodynamic proper-ties of the Shastry-Sutherland model with iPEPS, we obtainthe magnetic specific heat [Eq. (16)] from the numericalderivative of the energy as a function of the temperature
1Alternatively, one may also regard this iPEPS ansatz as a directrepresentation of a thermal density operator ρ(β/2) at inverse tem-perature β/2, where each site has both an incoming and an outgoingphysical index, and Fig. 4(e) corresponds to ρ(β ) = ρ†(β/2)ρ(β/2).
2Specifically, we employ a second-order Trotter-Suzuki decompo-sition in which the sequence of two-body time-evolution operators isreversed at every other time step.
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
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0.5
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0.7
0.8
0.08 0.09 0.1 0.11 0.12
0.6
0.65
0.7
0.75
0.2 0.25 0.3 0.35 0.40.34
0.36
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0.4
(a)
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.15 0.2 0.25 0.3 0.35
0.275
0.28
0.285
0.29
0.295
0.3
(b)
FIG. 5. Comparison of D = 14 iPEPS data for (a) the magneticspecific heat and (b) the magnetic susceptibility for the Shastry-Sutherland model with J/JD = 0.63. The reference line correspondsto a D = 14 simple-update calculation with time step τ = 0.04,boundary bond dimension χ = 50, and in (b) a field h/JD = 0.01.For each of the other lines, one parameter has been varied in thecalculation with respect to the reference line, as indicated in thelegend. The insets magnify the maxima and the minimum.
and the magnetic susceptibility [Eq. (17)] from the numericalderivative of the magnetization with respect to a small externalmagnetic field h. To illustrate the dependence of our results onthe parameters τ , D, and χ intrinsic to the iPEPS procedure,in Figs. 5 and 6 we show example data for the coupling ratioJ/JD = 0.63.
Our primary results are obtained using a second-orderTrotter-Suzuki decomposition with a time step τ = 0.04. Todemonstrate that this value is sufficiently small, meaning thatthe Trotter discretization error is negligible compared to thefinite-D errors (which we discuss below), in Fig. 5 we showthe comparison with results obtained using a time step τ =0.005. Further, it is easy to show that a relatively small bound-ary bond dimension, such as χ = 50 for D = 14, is largeenough for the accurate computation of expectation values. As
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0 0.1 0.2 0.3 0.4 0.50
0.1
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0.5
0.6
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0.8
0.08 0.09 0.1 0.11 0.12
0.6
0.65
0.7
0.75
0.2 0.25 0.3 0.35 0.40.34
0.36
0.38
0.4
(a)
0 0.2 0.4 0.6 0.8 10
0.05
0.1
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0.25
0.3
0.15 0.2 0.25 0.3 0.35
0.275
0.28
0.285
0.29
0.295
0.3
(b)
FIG. 6. iPEPS results for (a) the magnetic specific heat and(b) the magnetic susceptibility of the Shastry-Sutherland model withJ/JD = 0.63, obtained by the simple-update method for differentvalues of the bond dimension D. The insets magnify the maxima andthe minimum.
Figs. 5(a) and 5(b) make clear, results obtained using χ = 80are almost identical to those using χ = 50. To compute themagnetic susceptibility we use a field h/JD = 0.01, which issufficiently small that the discretization error is not signifi-cant; a comparison with h/JD = 0.005 is shown in Fig. 5(b).
As noted above, the key figure of merit in iPEPS calcula-tions of physical quantities is D. Because the “appropriate”value of D to capture the physics of a system varies widelywith the problem at hand, we will focus throughout ouranalysis of the Shastry-Sutherland model on benchmarkingour results by comparing D values. To begin our investigationof the D-dependence and finite-D convergence of our results,in Fig. 6 we present iPEPS data for J/JD = 0.63, obtainedby the simple-update method for a range of different bonddimensions. We find in Fig. 6(a) that the position of thespecific-heat peak shifts slightly (4%) downward in temper-ature with increasing D, while its height is also reduced(2%). However, these changes take place largely from D = 12to D = 16, whereas changes between the D = 16 and D =
18 data are scarcely discernible on the scale of the figure.In the susceptibility [Fig. 6(b)], finite-D effects are clearlyvery small for T/JD > 0.2; for 0.05 < T/JD < 0.2 a smallincrease can be observed with increasing D.
Finally, to investigate the difference between results byusing the simple- and full-update approaches, we return toFig. 5. At fixed D = 14, this difference is relatively smallin the specific heat, amounting to changes of less than 2%in the position and height of the peak [inset, Fig. 5(a)]. Forthe susceptibility, differences between the two approachesare minimal [Fig. 5(b)]. Because this difference is small incomparison with the finite-D effects shown in Fig. 6, we focushenceforth on the simple-update approach, which allows usto reach bond dimensions as large as D = 20 in the presentproblem, rather than the full-update approach, where D = 14represents a practical upper limit.
IV. THERMODYNAMICS OF THESHASTRY-SUTHERLAND MODEL
WITH 0.60 � J/JD � 0.66
We begin the systematic presentation and comparison ofour thermodynamic results by making contact with the previ-ous state of the art, as shown in Ref. [25]. The analysis of theseauthors, which included ED, QMC, and high-temperature se-ries expansions, terminated at the coupling ratio J/JD = 0.60,where all three methods were beyond their limits. However, itwas shown that QMC remains a highly accurate technique,with only minimal sign problems and thus applicable withlarge system sizes, for J/JD < 0.526. Thus we first show, inFig. 7, the magnetic specific heat and susceptibility for J/JD =0.50 computed by the TPQ method for system sizes N = 32and 36 and by the iPEPS method with D = 16 and 18. Bycomparison with the large-system QMC benchmark (N = 200sites), we observe that our iPEPS data, arguably a product ofthe most recent and least well-characterized technique, agreeprecisely for both bond dimensions shown.
For ED-based methods, it was known already [25] thatclusters of size N = 20 give an adequate account of the ther-modynamics at this coupling ratio, with the exception of thevery peak in C(T ) [Fig. 7(a)]. Nevertheless, our TPQ resultscontain two surprises, in the form of the rather large errors andthe significant difference between results for the two clustersizes. Regarding how accurately the TPQ method reproducesboth C(T ) and χ (T ) at temperatures around the C(T ) peak,we note that our N = 32 results do indeed lie close to theQMC benchmark for this system size, with the discrepancybeing well within the error tube. Quite generally, the originof these somewhat large errors lies in the very low density oflow-lying states in the spectrum, as pointed out in Ref. [28],and this is certainly the case in the Shastry-Sutherland modelwhen the gap is as large as its value at J/JD = 0.5 (moredetails may be found in Sec. V A). We will show that our TPQresults become progressively more accurate as the systemapproaches the QPT, which reduces the gap, until couplingratios very close to the transition. With regard to the effects ofthe shape or geometry of the cluster, regrettably the methodallows little control beyond the absolute value of N .
In Fig. 8 we move into the previously intractable parameterregime by showing the specific heat and susceptibility at
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0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.00
0.05
0.10
0.15
0.20
0.25
0.30
χJ
D
(b) J/JD = 0.50
ED N = 20
TPQ N = 32, R = 80
TPQ N = 36, R = 20
iPEPS D = 16
iPEPS D = 18
QMC N = 32
QMC N = 200
0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C
(a) J/JD = 0.50
ED N = 20
TPQ N = 32, R = 80
TPQ N = 36, R = 20
iPEPS D = 16
iPEPS D = 18
QMC N = 32
QMC N = 200
FIG. 7. (a) Magnetic specific heat of the Shastry-Sutherlandmodel at J/JD = 0.50, computed by the TPQ method with N = 32and 36 and by the iPEPS method with D = 16 and 18. Shownfor comparison are ED results with N = 20 and QMC results withN = 32 and 200. (b) Corresponding magnetic susceptibility.
J/JD = 0.60 computed by the TPQ method for system sizesN = 32 and 36 and by the iPEPS method with D = 14, 16,and 18. Focusing first on the specific heat [Fig. 8(a)], westill observe a single peak at low temperatures (T/JD � 0.12),but followed by a very flat plateau extending to T/JD �0.45. It is clear that our TPQ results, augmented by an EDcalculation with N = 28 reaching temperatures just above thepeak, exhibit finite-size effects in this region. Both the positionand the sharpness of the peak show a significant evolution asthe system size is augmented, although from N = 32 to 36 thedifference is within the error bars of the method. As expectedfrom Sec. III, the three iPEPS bond dimensions show goodconvergence, in fact to a peak shape exactly between the twoTPQ estimates. Outside the peak region, all methods and sizesagree extremely well and also converge at high temperaturesto the values obtained by QMC, which returns results ofacceptably low statistical uncertainty at T/JD � 0.25 for thiscoupling ratio [25].
The corresponding susceptibility [Fig. 8(b)] is less sensi-tive to differences in either system size or bond dimension.
0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
χJ
D
(b) J/JD = 0.60
ED N = 20
ED N = 28
TPQ N = 32, R = 80
TPQ N = 36, R = 20
iPEPS D = 14
iPEPS D = 16
iPEPS D = 18
QMC N = 32
0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C
(a) J/JD = 0.60
ED N = 20
ED N = 28
TPQ N = 32, R = 80
TPQ N = 36, R = 20
iPEPS D = 14
iPEPS D = 16
iPEPS D = 18
QMC N = 32
FIG. 8. (a) Magnetic specific heat of the Shastry-Sutherlandmodel at J/JD = 0.60, computed by the TPQ method with N = 32and 36 and by the iPEPS method with D = 14, 16, and 18. Shownfor comparison are ED results with N = 20 and with N = 28 attemperatures T/JD � 0.14, as well as QMC results with N = 32from Ref. [25]. (b) Corresponding magnetic susceptibility.
Indeed, only an ED calculation with N = 20 is not capableof capturing the maximum of χ (T ) for this coupling ratio.However, the two TPQ results do differ concerning the exactlocation in temperature of the rapid rise in χ (T ), and the fullyconvergent iPEPS results appear to offer the benchmark. Allof our calculations agree closely on the location of the broadmaximum. We comment that the characteristic temperature ofχ (T ) need not match that of C(T ), given that the former hascontributions only from magnetic states but the latter includesthe contributions of singlets. However, at this coupling ratio,where the one-triplon excitations of the system lie below alltwo-triplon bound states [25], the peak in C(T ) and the rise ofχ (T ) do indeed coincide. We discuss the excitation spectrumof the model in detail in Sec. V A.
As noted in Sec. I, the coupling ratio J/JD = 0.63 has spe-cial significance in the Shastry-Sutherland model because ofits proposed connection to the material SrCu2(BO3)2, and forthis reason we have already used it for benchmarking purposesin Secs. II and III. In Fig. 9 we compare our TPQ and iPEPS
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0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
χJ
D
(b) J/JD = 0.63
ED N = 20
ED N = 28
TPQ N = 32, R = 80
TPQ N = 36, R = 40
TPQ N = 40, R = 5
iPEPS
iPEPS
iPEPS
iPEPS
D = 14
iPEPS D = 16
iPEPS D = 18
QMC N = 32
FIG. 9. (a) Magnetic specific heat of the Shastry-Sutherlandmodel at J/JD = 0.63, computed by the TPQ method with N = 32,36, and 40 and by the iPEPS method with D = 14, 16, and 18. Shownfor comparison are ED results with N = 20 and with N = 28 attemperatures T/JD � 0.125, as well as QMC results with N = 32.(b) Corresponding magnetic susceptibility.
results for C(T ) and χ (T ). Independently of either method,we observe that the specific-heat peak has become sharper,although not any taller, and that the drop on its high-T side hasturned into a true minimum, before C(T ) recovers to a broadmaximum around T/JD = 0.45. The position of the sharppeak has moved down to T/JD � 0.10, which is a remarkablylarge shift for such a small change in coupling ratio. Althoughthe corresponding features are far less pronounced in χ (T ), itit clear that onset is considerably steeper than at J/JD = 0.60and that the temperature scales both for it and for the flatmaximum are lower.
Focusing on the details of C(T ) [Fig. 9(a)], once again ourTPQ results for N = 32 and 36, particularly in combinationwith N = 28 ED, show nontrivial changes around the peak,in both its position and its height. However, the N = 40 dataappear to offer a systematic interpolation of both quantities.In our iPEPS results, D = 14 no longer appears representativeof the large-D limit at this coupling ratio, but finite-D effectsseem to become very small at higher D. Qualitatively, our
0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C
(a) J/JD = 0.65
ED N = 20
ED N = 28
TPQ N = 32, R = 80
TPQ N = 36, R = 20
iPEPS D = 14
iPEPS D = 16
iPEPS D = 18
QMC N = 32
0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
χJ
D
(b) J/JD = 0.65
ED N = 20
ED N = 28
TPQ N = 32, R = 80
TPQ N = 36, R = 20
iPEPS D = 14
iPEPS D = 16
iPEPS D = 18
QMC N = 32
FIG. 10. (a) Magnetic specific heat of the Shastry-Sutherlandmodel at J/JD = 0.65, computed by the TPQ method with N = 32and 36 and by the iPEPS method with D = 14, 16, and 18. Shownfor comparison are ED results with N = 20 and with N = 28 attemperatures T/JD � 0.11, as well as QMC results with N = 32.(b) Corresponding magnetic susceptibility.
optimal iPEPS peak appears to be very close to the form atwhich one might expect the TPQ results to converge withsystem size, and we will quantify this statement below. Inχ (T ) [Fig. 9(b)], the two methods achieve full convergencefor all N and D, despite the lowering effective temperaturescales, and we draw attention to the fact that previous EDand QMC studies at this coupling ratio could not capture thislow-temperature behavior with any accuracy; we revisit thispoint in the context of SrCu2(BO3)2 in Sec. V C.
Proceeding yet closer to the QPT, Fig. 10 shows thespecific heat and susceptibility at the coupling ratio J/JD =0.65. Clearly, the “peak-dip-hump” shape of C(T ) has becomesignificantly more pronounced in all respects than at J/JD =0.63; the peak is sharper, the dip is deeper, and the separationof the peak and the broad hump has increased, with thepeak moving below T/JD = 0.09 and the high-T maximumcentered around T/JD = 0.5. Once again, the progressionfrom our ED data at N = 28 to the TPQ results with N = 32and then 36 shows an increasing ability to capture the tip
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0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
χJ
D
(b) J/JD = 0.66
ED N = 20
ED N = 28
TPQ N = 32, R = 80
TPQ N = 36, R = 20
iPEPS D = 14
iPEPS D = 16
iPEPS D = 18
0.0 0.1 0.2 0.3 0.4 0.5T/JD
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C
(a)
J/JD = 0.66
ED N = 20
ED N = 28
TPQ N = 32, R = 80
TPQ N = 36, R = 20
iPEPS D = 14
iPEPS D = 16
iPEPS D = 18
FIG. 11. (a) Magnetic specific heat of the Shastry-Sutherlandmodel at J/JD = 0.66, computed by the TPQ method with N = 32and 36 and by the iPEPS method with D = 14, 16, and 18. Shownfor comparison are ED results with N = 20 and with N = 28 at tem-peratures T/JD � 0.10. (b) Corresponding magnetic susceptibility.
of the peak, but it cannot be said that convergence has beenreached [inset, Fig. 10(a)]. As at J/JD = 0.63, our iPEPS datafor D = 16 and 18 are in essence indistinguishable and dosuggest the peak to which the TPQ results are converging.At all other parts of the C(T ) curve, convergence is completeby all methods. In the corresponding susceptibility, again theED and TPQ size sequence provides a systematic convergencetowards the “squareness” of the turnover from rapid rise tobroad maximum, while the higher iPEPS bond dimensionshave converged to the shape of this feature.
We take the coupling ratio J/JD = 0.66, shown in Fig. 11,as the upper limit to the validity of comparing our methods.Physically, it is obvious that the peak-dip-hump shape of C(T )becomes yet more pronounced, that the dip is deeper, and thatthe peak is even lower lying and sharper than at J/JD = 0.65[Fig. 11(a)]. Numerically, it appears that both our methodsshow slower convergence, not only around the peak but also inthe dip. In more detail, our N = 32 and 36 TPQ results and ourD = 16 and 18 iPEPS results both offer internally consistentpeak shapes. However, it is no longer clear that the TPQ and
iPEPS results might extrapolate to the same limit, althoughthis could be because the effective location of the QPT differsfor the two methods due to finite-D and -N corrections,implying closer proximity to a competing phase in one thanin the other. On the overall shape of C(T ), we commentthat the hump can be understood as the energy scale of localspin-flipping processes on the scale of JD, while the low-Tpeak is the focus of our discussion in Sec. V A. Concludingwith the susceptibility at J/JD = 0.66 [Fig. 11(b)], again ouriPEPS results offer a consistent picture of the very steep onsetand square maximum, while our TPQ results suggest betteragreement with those of iPEPS at N = 32 than at N = 36.
The results in Figs. 8–11 capture the thermodynamicproperties of the Shastry-Sutherland model with a degreeof precision that was entirely unattainable by all previoustechniques. It is safe to say that no prior numerical studieshad even identified the peak-dip-hump form of the purelymagnetic specific heat with any reliability, let alone givenan indication of how narrow the low-T peak becomes in theregime near the QPT. While the square shape of χ (T ) hasbeen known since the early measurements on SrCu2(BO3)2
[8,12], again no numerical approaches had reproduced thisform. We discuss both the physics underlying these featuresand the experimental comparison in Sec. V.
The thermodynamic properties of the Shastry-Sutherlandmodel nevertheless set an extremely challenging problem inthe regime close to the QPT, even in the dimer-product phasewhere the ground state is exact and no finite-temperaturetransition is expected. It is clear in Figs. 8–11 that neitherour TPQ nor our iPEPS calculations can be judged to befully convergent, and thus accurate, as J/JD approaches thepresumed critical value of 0.675. Because both are limitedby their control parameters, N for the TPQ method and Dfor the iPEPS method, an optimal reproduction of the exactthermodynamics would be obtained by extrapolating both setsof results to infinite N or D.
By inspection of the full peak shape in C(T ), our TPQresults can only be said to have reached convergence atN � 40 for J/JD � 0.63. Similarly, it is not clear that ouriPEPS results for the largest available D values (we haveperformed calculations with D = 18 at all temperatures andwith D = 20 for temperatures around the peak) can be saidto have converged to the shape of the peak at J/JD = 0.66.For extrapolation purposes, we first extract the primary char-acteristic features of this low-T peak, namely its position andheight. In Fig. 12 we show the convergence of both quantitiesas functions of 1/N in our ED and TPQ calculations and of1/D in our iPEPS calculations, for each of the coupling ratiosJ/JD = 0.60, 0.63, 0.65, and 0.66.
Focusing first on our iPEPS results, it is evident that thevalues we have obtained for both the position and the heightof the C(T ) peak show an excellent linear convergence with1/D for all coupling ratios. Both quantities decrease with in-creasing D and in fact their slopes are remarkably insensitiveto J/JD. Thus we estimate the putative infinite-D limit ofthe peak characteristics (shown by a filled triangle) by linearextrapolation in 1/D and we take the difference of this limitfrom the result at the largest computed D value as an estimateof the (one-sided) error bar. While all of the iPEPS estimatesdecrease monotonically with increasing D, in the large-D limit
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0 0.04 0.08 0.121/D or 1/N
0.4
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Cm
ax
iPEPSEDTPQ
0 0.04 0.08 0.121/D or 1/N
0.1
0.11
0.12
0.13
0.14
0.15
TC m
ax
iPEPSEDTPQ
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Cm
ax
iPEPSEDTPQ 0.07
0.08
0.09
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TC m
ax
iPEPSEDTPQ
0.4
0.5
0.6
0.7
0.8
Cm
ax
iPEPSEDTPQ
0.06
0.07
0.08
0.09
0.1
0.11
TC m
ax
iPEPSEDTPQ
0 0.04 0.08 0.121/D or 1/N
0.4
0.5
0.6
0.7
0.8
Cm
ax
iPEPSEDTPQ
0 0.04 0.08 0.121/D or 1/N
0.06
0.07
0.08
0.09
0.1
TC m
ax
iPEPSEDTPQ
J / JD
= 0.60
(a)
J / JD
= 0.60(b)
J / JD
= 0.63
(c)
J / JD
= 0.63(d)
J / JD
= 0.65
(e)
J / JD
= 0.65(f)
J / JD
= 0.66
(g) J / JD
= 0.66
(h)
FIG. 12. Characteristic position and height of the low-T peak inthe magnetic specific heat of the Shastry-Sutherland model, shownfor our ED and TPQ results as functions of 1/N and for our iPEPSresults as functions of 1/D, with (a) and (b) J/JD = 0.60, (c) and(d) J/JD = 0.63, (e) and (f) J/JD = 0.65, and (g) and (h) J/JD =0.66.
one does anticipate exponential convergence in D rather thanlinear convergence in 1/D. Thus we expect the true D = ∞limit to lie within the interval indicated by the error bar.
Turning to the ED and TPQ results in Fig. 12, the peakheights at all coupling ratios show a quite systematic conver-gence, from below and with changing slopes, towards valuesfully consistent with our iPEPS estimates. By contrast, thefinite-size evolution of the peak positions is less clear, butit is true to state for every coupling ratio that their valuesfor all system sizes are either clustered closely around theextrapolated iPEPS values or are trending towards these.
Thus we may conclude that our TPQ and iPEPS resultsare consistent and convergent for even the most challeng-ing features of the most challenging region of the Shastry-Sutherland phase diagram. As such they may be used as areliable statement of the excitation spectrum as a functionof the coupling ratio, which is the topic of Sec. V A. Wediscuss the implications of the convergence details in Fig. 12for future numerical development in Sec. V B.
V. DISCUSSION
A. Spectrum of the Shastry-Sutherland model
In Fig. 2(a) we showed our collected results for the specificheat of the Shastry-Sutherland model across the range of
coupling ratios 0.60 � J/JD � 0.66. As the system ap-proaches the QPT from the dimer-product to the plaquettestate, one observes the systematic emergence of a narrow low-energy peak, followed at higher temperatures by a deepeningdip before a second broad maximum on the scale of JD/2.The energy scale of the concentration of low-lying states con-tributing to the peak falls steeply as the QPT is approached.From the corresponding susceptibilities, shown in Fig. 2(b), itis clear that the decreasing energy scale is largely similar fortriplet (and higher spinful) excitations, and hence at minimumthat it is not a special property of singlets; the physics of theapproach to the QPT seems to affect all excitations in the samegeneral manner.
Before discussing the nature of the low-lying spectrum, wecharacterize the height and temperature scale of the specific-heat peak, and the temperature scale of the susceptibilityonset, as the coupling ratio approaches the QPT. The peakposition T C
max, shown in Fig. 13(a), clearly falls faster thanlinearly in J/JD, but does not approach zero at the QPT. Basedon the expectation that the peak position as a function of J/JD
should follow the gap �m of the system, in Fig. 13(a) we showthis quantity based on ED data from the N = 36 cluster [25]and with a multiplicative prefactor of 0.28. The fact that thesefalling energy scales do not vanish at the QPT is consistentwith the first-order nature of this transition [5]. The behaviorof the peak height [Fig. 13(b)] is less evident, in that it fallson approach to the QPT in our TPQ calculations but increasesin our iPEPS ones, the rate of change accelerating with J/JD
in both cases. Given the strong finite-size effects visible inour ED and TPQ calculations for J/JD > 0.63 (Fig. 12), hereit appears that our iPEPS results are more representative ofthe large-N limit. The final piece of information one maywish to extract about the peak is its width, for which we showin addition the temperature T C
1/2, at which the specific heatattains half its peak height [Fig. 13(a)]; because of the dip andhump on the high-T side, we focus only on the low-T side. Weobtain a characteristic width from the quantity T C
max − T C1/2,
shown as the shaded gray region. We observe that the half-height and full-height positions, and hence the width, scale toa good approximation in the same way with J/JD, separatedonly by constant factors, and conclude that all the peak shapesare actually self-similar on the low-temperature side, withonly one characteristic J/JD-dependent energy scale.
Because the maximum of χ (T ) is broad and flat, for anaccurate characterization of the susceptibility we focus noton its position T χ
max but on the temperature T χ
1/2 [79], atwhich χ (T ) reaches half its maximum value during its rapidonset. Both quantities are shown in Fig. 13(c), which indeedconfirms significantly larger uncertainties in T χ
max than in T χ
1/2.Throughout the range of Fig. 13, T χ
1/2 tracks both T Cmax and
T C1/2 rather closely, albeit with a slightly slower decline very
near the QPT. In this sense it functions as a comparablediagnostic of the falling energy scale in the system. Followingthe expectation that χ (T ) depends on the gap to spinfulexcitations, in Fig. 13(c) we compare this scale with thetriplet gap �T (J/JD), finding a constant of proportionality ofapproximately 0.22.
Figure 13 is a very specific diagnostic for the nature of thespectrum of low-lying energy levels, reflecting in particular
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ALEXANDER WIETEK et al. PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66
J / JD
0
0.05
0.1
0.15
0.2
TC m
ax /
J D, T
C 1/2 /
J D
cΔm
, N = 36
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Cm
ax
ED, N = 20ED, N = 24ED, N = 28TPQ, N = 32TPQ, N = 36TPQ, N = 40QMC, N = 200iPEPS, extrapolated
0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66J / J
D
0
0.1
0.2
0.3
0.4
Tχ m
ax /
J D, T
χ 1/2 /
J D
dΔT, N = 36
(a)
(b)
(c)T
χmax
Tχ1/2
TC
max
TC
1/2
FIG. 13. (a) Position T Cmax and (b) height Cmax of the peak in
the specific heat of the Shastry-Sutherland model in the regime ofcoupling ratios 0.50 � J/JD � 0.66, as determined from our ED,TPQ, and iPEPS calculations. Also shown in (a) is the half-heightT C
1/2 on the low side of the peak, from which one may deducean effective peak width (shaded gray region). The iPEPS error-barconventions are as in Fig. 12. At J/JD = 0.50 we compare in additionwith the large-N QMC result. Shown for reference in (a) is thefunction c �m(J/JD ), where �m = min[�S, �T ], with �S and �T ,respectively, the gaps of the lowest singlet and triplet excitationsobtained for a system of size N = 36, is the minimal gap in thesystem; the constant of proportionality is c = 0.28. (c) Positionsof the peak T χ
max and of the half-height T χ
1/2 in the susceptibility ofthe Shastry-Sutherland model in the same range of coupling ratios.Again the shaded region may be considered as an effective width.Shown for reference is the function d �T , again obtained for N = 36,with constant of proportionality d = 0.22.
0.0 0.1 0.2 0.3 0.4 0.5 0.6
J/JD
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
ΔS
ΔT
2ΔT
ΔT + ΔS
2ΔS
S = 1 states
S = 0 states
4.0
E/J
D
N = 20
N = 24
N = 28
FIG. 14. Low-lying energy spectrum of the Shastry-Sutherlandmodel in the S = 0 and S = 1 sectors, computed by ED using clustersof sizes N = 28 (to E = 2.6JD), N = 24 (to 3.75JD), and N = 20(to 4.2JD) and shown over the full range of coupling ratios J/JD
in the dimer-product phase. The excitation energy is defined byE = E − EGS, where EGS is the energy of the ground state. Theshading represents regions with discrete but dense singlet (blue) andtriplet excitations (pink). The lowest pink line shows the one-triplonsector, whose lower edge is �T (J/JD ); the yellow line shows thetwo-triplon scattering threshold 2�T (J/JD ). The lower edge of thelower blue sector, which consists of singlet bound states of twotriplons, is �S (J/JD ). We draw attention to how few states in thethree-triplon sector fall below the green line (�S + �T ) and in thefour-triplon sector below the black line (2�S).
a large number of states that become more nearly degenerateand are characterized by an ever-smaller (but always finite)gap as the system approaches the QPT. Reference [25] showeda schematic illustration of the energy spectrum of the Shastry-Sutherland model as a function of the coupling ratio J/JD,based on Lanczos ED calculations to obtain the lowest levelsof a cluster of size N = 36 sites. For more specific insight intothe nature of the states revealed by the Lanczos procedure,here we have computed a much larger number of low-energystates, but for smaller clusters (of sizes up to N = 28). InFig. 14 we gather these results to show the low-lying spectrumof the system in the S = 0 and S = 1 sectors over the fullrange of coupling ratios. We stress that, given the largenumber of states in these calculations, it is not easy to observethe discrete structure of the spectrum even in its lowest-lyingregions, and as in Ref. [25] we use shading to indicate thesupport of these excitations.
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THERMODYNAMIC PROPERTIES OF THE … PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66J / J
D
0
20
40
60
80
100
120
140
No.
of
boun
d st
ates
S = 0S = 1
FIG. 15. Numbers of low-lying singlet and triplet states on anN = 28 lattice, shown as functions of J/JD in the dimer-productphase. Symbols show ED data and connecting lines are guides tothe eye. The horizontal green line shows the number of independenttwo-triplon scattering states.
In Fig. 14 the thin red line at the bottom of the triplet spec-trum is the elementary triplon excitation, which has energyE = JD at J = 0 and remains extremely narrow (nondisper-sive) throughout the dimer-product phase due to the perfectfrustration (Fig. 1). It is clear that significant numbers of two-triplon and other multitriplon states, in both the spin sectorsshown, move to very low energies as the system approachesthe QPT. Because the lowest states in the singlet sector mustbe of two-triplon origin and these cross below the one-triplonenergy around J/JD = 0.60 (the exact value depends on N),it is equally clear that the physics of the Shastry-Sutherlandmodel in the regime near the QPT is dominated by states onemay classify as “strongly bound”. To specify the meaningof this term, the yellow line shows the quantity 2�T (J/JD),i.e., twice the single-particle gap, which corresponds to thetwo-triplon scattering threshold, and near the QPT many stateslie significantly below this energy. We comment that theseparate “families” of excitations that can be traced back toindependent two-triplon states at energy E = 2JD at J = 0,and similarly for higher triplon numbers (Fig. 14), exhibita much larger dispersion across the Brillouin zone than dothe one-triplon states, due to the smaller effects of geometricfrustration on the propagation of bound states [4,80,81]. Al-though some authors have performed perturbative studies ofthe lowest-lying bound states of the Shastry-Sutherland model[80], we are not aware of previous attempts to count them sys-tematically for comparison with thermodynamic properties, aswe do next.
Beyond illustrating significant binding-energy effects inthe two- and higher-triplon sectors, Fig. 14 does not providefurther information about the structure of the spectrum. As astep in this direction, Fig. 15 shows the numbers of singlet andtriplet states lying below the threshold given by the yellow linein Fig. 14. The number of triplet states excludes the N/2 statesof the one-triplon band. For comparison, the expected numberof two-triplon scattering states 1
4 N ( 12 N − 1) is shown by the
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
20
40
60
80
100
No.
of
stat
es
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
20
40
60
80
100
120
No.
of
stat
es
S = 1, N = 16S = 1, N = 20S = 1, N = 24S = 1, N = 28S = 0, N = 16S = 0, N = 20S = 0, N = 24S = 0, N = 28
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ~E / J
D
0
20
40
60
80
100
120
140
160
No.
of
stat
es
J / JD
= 0.60(a)
J / JD
= 0.63(b)
J / JD
= 0.66(c)
FIG. 16. Total number of excited states lying below the energy Eat coupling ratios (a) J/JD = 0.60, (b) J/JD = 0.63, and (c) J/JD =0.66. Blue lines indicate S = 0 excitations, red lines S = 1. Increas-ing line widths indicate increasing system sizes N on which EDcalculations were performed. Vertical green lines denote the two-triplon scattering thresholds for the corresponding system sizes.
green line. Clearly, the numbers of “bound” states in bothsectors are significant fractions of the number of scatteringstates throughout the region J/JD � 0.55. Further, the numberof S = 1 states is comparable to that in the S = 0 sector,despite the fact that the energy gain is greater in the latter.As the QPT is approached, specifically, once J/JD > 0.62for N = 28 [Fig. 15], the numbers of low-lying states inboth sectors increase well beyond the number of independenttwo-triplon states.
Before discussing the nature of these states, for furtherinformation about the low-lying spectra we show in Fig. 16the total numbers of singlet (blue) and triplet states (red) lyingbelow a given energy E in ED calculations performed with
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ALEXANDER WIETEK et al. PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
four different system sizes for three fixed values of J/JD in theinterval near the QPT. Here the N/2 one-triplon excitations areincluded in the triplet count, appearing as the plateau regionin all the red lines directly above their onset. We note that allexcitations with higher spin, S � 2, lie above the two-triplonscattering threshold for all J/JD values shown; while bindingeffects have also been observed in the S = 2 sector [82], theseare extremely small and are not detectable in clusters up toN = 28 due to finite-size effects.
The evolution of this spectrum as a function of the cou-pling ratio contains two additional messages beyond the ever-increasing number of very-low-lying states shown in Fig. 15.First, the steadily decreasing gap in the one-triplon sector canbe expected to control the response to �S �= 0 processes overmost of regime 0.60 � J/JD � 0.66, but very near the QPT itis overwhelmed by the high density of other triplet states thatfinally fall to this gap. Second, the dominant feature of thespectrum in this regime is the large number of low-lying sin-glet states, which are governed by an energy scale that in turnfalls significantly more quickly with J/JD than the triplet gap.
To discuss the nature of these states, and hence the con-sequences of our calculated spectra for the thermodynamicresponse functions computed in Sec. IV, it is helpful tocompare and contrast the Shastry-Sutherland model with thefully frustrated two-leg spin ladder. In this 1D model one alsofinds a striking thermodynamic response at low temperatures[79,83], which can be traced to a proliferation of low-energymultimagnon bound states upon approaching a first-orderQPT. While it is clear that the proliferation of low-lying statesis the origin of the low-temperature properties of the Shastry-Sutherland model, there are some important differences be-tween the 1D and 2D systems, most notably concerning themechanism behind the formation of these states.
First, in the fully frustrated two-leg ladder, it was clearthat these states were truly bound, meaning that they wereintrinsic combinations of n contributing triplons, with n �2 becoming very large near the QPT. By contrast, in theShastry-Sutherland system one may identify essentially allof the additional low-lying states, whose origin clearly liesin n � 3 triplons, with the expected scattering states of twosinglet two-triplon bound states or of a singlet bound stateand an elementary triplon. It is self-evident that when theenergy of a singlet bound state of two triplons falls beyond theone-triplon gap, the lowest-lying scattering states of a pair ofsinglet bound states must fall below the two-triplon scatteringthreshold in the S = 0 sector, shown by the black line inFig. 14, and so does the scattering state of a singlet boundstate and an elementary triplon in the S = 1 sector, shownby the green line in the same figure. Thus the proliferationof low-lying excitations in the regime J/JD > 0.6 can, due tothe differences in state counting in two dimensions, be driven(almost) completely by the strength of the two-triplon bindingeffect in the singlet sector. It is for this reason that we do notrefer to all of the low-lying states we observe here as “boundstates”. Concerning the possibility of true multitriplon bindingeffects, as noted above we may comment only that, if present,these are sufficiently weak that we are unable to detect them.
Second, at the QPT in the fully frustrated ladder, thebound states cluster strongly at a single energy value ofapproximately 85% of the one-triplon gap [79,83], but in the
Shastry-Sutherland model we observe no such sharp struc-tures in the density of states. Specifically, although we findfor J/JD = 0.66 that low-lying singlet states far outnumberthe triplets at any energy E < 0.5JD [Fig. 16(c)], there is nosign of a discrete set of degenerate levels, at any energy, thatcould be related directly to the specific-heat peak. Instead, thenumber of excited states continues to rise with E , in fact at anincreasing rate, such that some very high densities of statesmay be found at energies of order 0.5JD above the lowestones in each panel of Fig. 16. The distinguishing feature ofthese broadly distributed states, at least in the singlet sector,is that their binding effects are energetically stronger than inthe 1D system. Although finite-size effects prevent us fromreaching the QPT (J/JD = 0.675), at J/JD = 0.66 the singlettwo-triplon bound states have already descended to only 54%of the one-triplon gap (on the N = 28 cluster). Thus thelow-temperature peak in C(T ) observed in Sec. IV emergesat a relatively lower energy scale than in the fully frustratedladder. Certainly the possibility of ascribing our results toonly one energy scale is consistent with our discovery inFig. 13(a) that the peak shapes are identical. In this regard,the strikingly narrow specific-heat peaks near the QPT arenot in fact anomalous, in that their relative width is constantand their appearance is a consequence of this one decreasingenergy scale. Regarding the susceptibility, the decrease ofthe one-triplon energy scale in Fig. 16 appears to provide areasonable account of the decrease in the “onset” temperatureindicator T χ
1/2 [Fig. 13(c)] over the entire range of J/JD, whilethe steep fall in energy of the other low-lying triplet states maybe responsible for the increasingly abrupt (square) turnover atthe maximum in χ (T ).
In summary, our detailed ED investigation of the low-energy spectrum (Figs. 14–16) reveals the rapid descent inenergy of a multitude of singlet and triplet states as the QPT isapproached. Despite the narrow nature of the C(T ) peak, thesestates do not converge to a single characteristic energy in ei-ther sector. In contrast to the n-triplon bound states of the fullyfrustrated ladder near its QPT, the energetics of the low-lyingstates of the Shastry-Sutherland model are dominated by thebinding energy of two triplons into a net singlet. Because thislocal bound state involves two neighboring orthogonal dimers[80,81], it has little or no relation with the plaquette singletsthat control the physics on the other side of the QPT (Fig. 1),and hence one may conclude that the first-order transitioninvolves a complete rearrangement of spin correlations in thesystem.
B. Numerical methods
We comment only briefly on the limitations and prospectsfor improvement of our numerical techniques. In a sense theShastry-Sutherland model near the QPT presents an ideal testbed for any method, in that the low-T peak in the specific heatbecomes progressively narrower, and thus more challengingto reproduce, as the QPT is approached. In our ED and TPQcalculations, it is both clear and completely unsurprising thatlarger cluster sizes improve the ability of both methods tocapture this peak (and the corresponding susceptibility shoul-der). Whereas the thermodynamic response at coupling ratioJ/JD = 0.60 could not be reproduced with complete accuracy
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THERMODYNAMIC PROPERTIES OF THE … PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
by N = 20 ED, or indeed N = 28 Lanczos ED (Fig. 8), wemay assert that J/JD = 0.63 can be treated accurately by theTPQ method with N � 40 (Fig. 9).
However, the limits to the TPQ method appear to bereached at J/JD > 0.65, where our calculations are no longerable to capture the anomalously narrow C(T ) peak that formsin this regime (Fig. 11). With reference to our discussion ofSec. V A, one may only speculate that the physics of this peakresides in scattering states of the singlet two-triplon boundstates whose fully developed real-space extent begins to ex-ceed this cluster size very close to the QPT. As noted in Sec. II,reaching larger cluster sizes with acceptable statistical errorsis not possible with the present computing power. Similarly,the TPQ method, like ED, is not limited by algorithmic com-plexity, and indeed our present calculations already exploit allavailable spin and spatial (cluster) symmetries, implying thatmore complex Hamiltonians would be subject to stricter limitson N .
By contrast, our iPEPS calculations have relatively modestCPU and memory requirements. To date they have been runonly on a single node, and thus larger-scale computationswould certainly become feasible through the future devel-opment of a highly parallelized code. In comparison withmore mature numerical methods, tensor-network calculationsremain in their relative infancy and as a result retain signif-icant scope for algorithmic improvement. In current imple-mentations, both CPU and memory requirements increase ashigh powers of the bond dimension D; although most devel-opments focus on reducing these exponents, it is not provenbeyond doubt that large D is the only route to an accuratetensor-network description. As one example, we commentthat higher accuracies can be achieved for the same D valuewithin our present calculations by the use of disentanglinggates acting on the ancillas [36,84].
Nevertheless, as demonstrated in Sec. III, the influenceof the other variables in the iPEPS method (Trotter step,CTM boundary bond dimension, and type of update) is smallin comparison with the effects of D. Physically, there is atpresent no known means of relating D as a figure of merit infinite-T (i.e., thermal-state purification) iPEPS implementa-tions with D in a ground-state iPEPS calculation. As with theproblem of which absolute value of D is able to capture thephysics of any given Hamiltonian (Sec. III), the most reliableapproach to date is the empirical one of increasing D andmonitoring changes and convergence. As we have shown inSecs. III and IV (Fig. 12), for the present problem we haveindeed reached a well-defined limiting regime of D, evenat J/JD = 0.66. Thus one may conclude that the Shastry-Sutherland model does not present a physical system wherecalculations are limited by the ability to capture long-rangeentanglement, as may perhaps be expected when the groundstate is a product state, all the excitations are gapped, and thenearby QPT is first-order.
C. Experiment
In the light of the data and insight obtained from applyingour two advanced numerical methods to the thermodynamicsof the Shastry-Sutherland model, we revisit the experimentalsituation in SrCu2(BO3)2. We note that low-lying singletexcitations in SrCu2(BO3)2 were documented by Raman scat-
tering soon after the discovery of the material [85] and laterthat the same method was used to study the low-lying spec-trum in an applied magnetic field [86]. The triplet excitationswere studied at first by electron-spin resonance (ESR) [87]and inelastic neutron scattering [10]. Subsequent high-fieldESR measurements were used to demonstrate the presenceof two-triplon bound states [88], while a recent applicationof modern neutron spectroscopy instrumentation studied thelowest bound triplet mode, and its response in a magneticfield, in detail [89]. The evolution of the triplet spectrum hasbeen studied under pressure [23], which as noted in Sec. Iacts to increase the ratio J/JD through the first-order QPT outof the dimer-product phase. Although an analogous study ofthe thermodynamic properties has appeared in parallel withthe completion of the present study [24], only one data setfor C(T )/T is shown at a pressure between ambient and theQPT; the low-T peak moves down by approximately 25% atthis pressure and undergoes a definite increase in sharpness.Around the QPT, extremely sharp C(T )/T peaks are foundin some, but not all, samples. We observe that the numericalmodeling that accompanies these measurements does notadvance the state of the art beyond that of Ref. [26].
We comment first that a rather accurate understandingof the high-temperature, and indeed high-field, properties ofSrCu2(BO3)2 can be obtained using the Shastry-Sutherlandmodel with a coupling ratio close to J/JD = 0.63, and it isthe remaining mystery surrounding the low-T response thatwe address here. We use our D = 18 iPEPS results as aconsistent indicator of the evolution of the specific heat andsusceptibility with the coupling ratio, and in Fig. 17 we showboth quantities for all J/JD = 0.60, 0.61, . . . , 0.66. For thespecific heat [Fig. 17(a)] we compare our results with the dataof Ref. [13] for SrCu2(BO3)2, from which we have subtracteda phonon contribution Cph = γ T 3 with γ = 0.5 mJ/(mol K4).For the susceptibility [Fig. 17(b)] we compare our resultswith the data of Ref. [12], using a g factor of gc = 2.28 [87].The optimal values of JD required to reproduce the positionand shape of the low-T peak in C(T ) and to reproduce therapid upturn and maximum height of χ (T ) are determinedseparately; in both cases they turn out to be strong functionsof the coupling ratio, and hence can be used for accurateestimation of this energy scale.
In Fig. 17(a) it is safe to say that the best account of thelow-T peak in C(T ) is given by J/JD = 0.62, with JD = 77 K.Somewhat surprisingly, it is not necessary to exploit to itslimits our abilities to resolve a very narrow peak in C(T ),as would be the case at higher coupling ratios. Because JD
is constrained by many other aspects of the experimental datafor SrCu2(BO3)2, its strong dependence on J/JD serves as astrict constraint on the coupling ratio. Turning to Fig. 17(b),the best account of χ (T ) is given by J/JD = 0.64, withJD = 91 K. Here it is safe to say that the fits in Figs. 17(a)and 17(b) are difficult to reconcile, certainly at the experi-mentally determined value of gc, and as a result we can con-clude that we have found the limits of the Shastry-Sutherlandmodel when applied to the SrCu2(BO3)2 problem. It is knownthat the appropriate spin Hamiltonian for this material con-tains additional terms, most notably an interlayer coupling,whose magnitude has been estimated at 9% of JD [4], andDzyaloshinskii-Moriya interactions both within and normal
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ALEXANDER WIETEK et al. PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
0 5 10 15 20
T [K]
0
1
2
3
4
5
6
C[J
/Km
ol]
(a)experimentJ/JD = 0.60, JD = 67 KJ/JD = 0.61, JD = 72 KJ/JD = 0.62, JD = 77 K
J/JD = 0.63, JD = 84 KJ/JD = 0.64, JD = 93 KJ/JD = 0.65, JD = 104 KJ/JD = 0.66, JD = 121 K
J/JDJ = 0.63, JDJ = 84 KJ/JDJ = 0.64, JDJ = 93 KJ/JDJ = 0.65, JDJ = 104 KJ/JDJ = 0.66, JDJ = 121 K
J/JD = 0.63, JD = 84 KJ/JD = 0.64, JD = 93 KJ/JD = 0.65, JD = 104 KJ/JD = 0.66, JD = 121 K
FIG. 17. (a) Magnetic specific heat and (b) susceptibility com-puted by iPEPS with D = 18 for coupling ratios 0.60, 0.61, . . . , 0.66and for JD values dictated by experiment. Specific-heat data in(a) were taken from Ref. [13] and a phonon contribution of0.5T 3 mJ/(mol K4) was subtracted. Susceptibility data in (b) weretaken from Ref. [12] and a g factor of 2.28 was used.
to the plane of the system, whose magnitudes have beenestimated [88] for both components at 3% of JD. Because it isnot the purpose of our present analysis to address these details,for further comparison in the present context we restrict ourattention to establishing the optimal Shastry-Sutherland fitfor SrCu2(BO3)2, to which end we adopt the compromisecoupling ratio J/JD = 0.63 and find that the correspondingoptimal JD is 87 K.
In Fig. 18 we show a quantitative comparison betweenthe results of Sec. IV, specifically Fig. 9, and the C(T ) dataof Ref. [13]. We observe that our best TPQ results (N =40) and our iPEPS results separately provide quantitativelyexcellent fits to the measured data. Our conclusion that sucha fit is optimized with the coupling ratio J/JD = 0.63 is inaccordance with the suggestion, made on the basis of small-system ED calculations, of Ref. [26]. Further, our determina-tion of the coupling constant, JD = 87 K, from the precisely
0 5 10 15 20
T [K]
0
1
2
3
4
5
6
C[J
/Km
ol]
J/JD = 0.63 experimentiPEPS D = 18ED N = 20TPQ N = 36, R = 40TPQ N = 40, R = 5
FIG. 18. Comparison of experimental measurements of the mag-netic specific heat, taken from Ref. [13] and with a phonon con-tribution 0.5T 3 mJ/(mol K4) subtracted, to numerical calculationsperformed by ED, TPQ, and D = 18 iPEPS methods for J/JD = 0.63with JD = 87 K.
known position T Cmax [Fig. 17(a)] of the C(T ) peak is also close
to the parameters deduced 20 years ago. As noted above, whilethese constants may also be constrained from high-T andmagnetization information, our results demonstrate finallythat they are consistent with the hitherto mysterious low-Tbehavior.
We stress that, despite reaching the previously unattainablegoal of reproducing the true shape of the low-T peak in C(T ),indeed for coupling ratios even higher than J/JD = 0.63,we do not find that a sharper peak is required to explainexperiment (our results at J/JD � 0.64 tend to exceed theexperimental peak height). Thus we deduce that, to the extentthat the Shastry-Sutherland model offers an acceptable ac-count of the physics of SrCu2(BO3)2 (i.e., to the extent that 3Dand Dzyaloshinskii-Moriya interactions may be neglected),the coupling ratio is J/JD = 0.63 ± 0.01. We have basedthis estimate on reproducing the shape of the low-T peak inC(T ); if we consider the dip region at higher temperatures,Fig. 18 shows a significant discrepancy between experimentand our best peak fits, but because the question of subtractingthe phonon contribution becomes more important here it isdifficult to ascribe the same quantitative reliability to our fits.
In Fig. 19 we perform the same exercise by showing aquantitative comparison between our results (Sec. IV) andthe data of Ref. [12] for the magnetic susceptibility ofSrCu2(BO3)2. Again we use the value JD = 87 K along withthe g factor gc = 2.28 noted above. We show the entiretemperature range in Fig. 19(a), which emphasizes the veryabrupt peak and the reliability of our model at all hightemperatures. In Fig. 19(b) we focus on the low-temperatureregime, where again the half-height T χ
1/2 admits only a rathernarrow range of coupling ratios. Thus our improved computa-tional methods confirm both the extent to which SrCu2(BO3)2
can be approximated by a pure Shastry-Sutherland modeland the efficacy of the parameter estimates made two
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THERMODYNAMIC PROPERTIES OF THE … PHYSICAL REVIEW RESEARCH 1, 033038 (2019)
iPEPS
iPEPS
FIG. 19. (a) Comparison of experimental measurements of themagnetic susceptibility, taken from Ref. [12], to numerical calcu-lations performed by ED, TPQ, and D = 18 iPEPS methods forJ/JD = 0.63. We use g = 2.28 and JD = 87 K. (b) Detail of thecomparison at low temperatures.
decades ago on the basis of significantly inferior numericaltechnology.
VI. SUMMARY AND PERSPECTIVES
We have introduced two advanced numerical techniques,the methods of typical pure quantum states and infiniteprojected entangled-pair states, for computing the thermo-dynamic properties of quantum magnets. Both approachesare unaffected by frustration and we apply them to computethe magnetic specific heat and susceptibility of the Shastry-Sutherland model in its dimer-product phase near the first-order quantum phase transition (QPT) to a plaquette phase.This challenging region of coupling ratios has remained en-tirely impervious to meaningful analysis by any previousnumerical techniques, and now we are able to reveal why. The
specific heat develops an increasingly narrow low-temperaturepeak, which is separated from the broad hump due to localprocesses by an emerging dip. This peak is caused by aproliferation of low-lying energy levels whose origin lies inscattering states of multiple two-triplon bound states, and itsanomalously low energy scale moves ever lower (albeit notto zero) as the QPT is approached. The analogous featurein the magnetic susceptibility is an increasingly rapid rise atan ever-lower temperature, followed by an abrupt maximum.This physics is a consequence of both high frustration andthe proximity to a QPT, and its manifestations have beendiscussed previously in highly frustrated 1D systems, but nowwe have captured its realization in two dimensions.
Our methods and benchmarks offer very wide scope forimmediate application. The list of open problems in frus-trated quantum magnetism is long, and we await with in-terest definitive results, meaning for extended or spatiallyinfinite systems, for the thermodynamic response of thetriangular, kagome, and J1-J2 square lattices. The abilityto fingerprint the energy spectrum at the level of ther-modynamic properties will provide a dimension of physi-cal understanding, and in some cases may even yield fur-ther insight into the ground state. Away from the classicproblems of geometrical frustration in Heisenberg models,there is a pressing need for thermodynamic information tocharacterize many other types of frustrated system, includ-ing spin-ice materials, candidate Kitaev materials, coupledanisotropic (Ising and XY ) chain and planar materials, candi-date chiral spin liquids, materials with Dzyaloshinskii-Moriyainteractions, topological magnets, and certain skyrmionsystems.
Technically, neither of our methods presents any barrierto the application of a magnetic field, which could be usedto control the relative singlet and triplet gaps and createqualitative changes in the thermodynamic response in manyof the above types of system. For the SrCu2(BO3)2 problem,the next step enabled by the capabilities developed here isclearly to follow the evolution of the system to and throughthe pressure-induced QPT into the plaquette phase [23,24].This process presents a clear need for the investigation ofthermodynamic properties, even if the physics of the materialbeyond the dimer-product phase is critically dependent onadditional Hamiltonian terms beyond the Shastry-Sutherlandmodel.
ACKNOWLEDGMENTS
We thank S. Capponi, P. Lemmens, J. Richter, Ch. Rüegg,J. Schnack, R. Steinigeweg, A. Tsirlin, and B. Wehinger forhelpful discussions. This work was supported by the DeutscheForschungsgemeinschaft under Grants No. FOR1807 and No.RTG 1995, by the Swiss National Science Foundation, and bythe European Research Council under the EU Horizon 2020research and innovation program (Grant No. 677061). TheFlatiron Institute is a division of the Simons Foundation. S.W.thanks the IT Center at RWTH Aachen University and the JSCJülich for access to computing time through JARA-HPC. A.H.is grateful for the use of the osaka cluster at the Université deCergy-Pontoise.
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